Factorizing the position-space photon propagator in QED corrections to lattice QCD correlators

This paper addresses the computational challenge of evaluating electromagnetic corrections to lattice QCD correlators by introducing and comparing factorization techniques for the position-space photon propagator that reduce the volume-squared complexity of the required sums, with applications demonstrated for hadronic vacuum polarization and the hadronic light-by-light contribution to the muon $(g-2)$.

The Gist

Imagine you are trying to understand the behavior of a tiny, invisible particle called a muon (a heavy cousin of the electron). Scientists want to know exactly how much this muon "wobbles" when it spins. This wobble, known as the anomalous magnetic moment, is like a fingerprint of the universe. If the fingerprint matches our current theories perfectly, great! If it doesn't, it means there are new, undiscovered forces or particles hiding in the shadows.

To get this fingerprint right, scientists use supercomputers to simulate the universe on a giant grid (a "lattice"). They are trying to calculate a specific correction caused by electromagnetism (light and electricity) interfering with the strong nuclear force.

The Problem: The "Infinite" Crowd

The main challenge in this paper is a computational nightmare. To calculate this correction, the computer needs to sum up the interactions of photons (particles of light) between two points inside the simulation.

Think of the simulation as a giant, 4-dimensional city. To get the answer, the computer has to ask: "If I place a photon here, and another photon there, what happens?"

The problem is that the city is huge. If you try to check every possible pair of locations for the two photons, the number of calculations explodes. It's like trying to introduce every person in a stadium to every other person in the stadium. If the stadium has 10,000 people, that's 100 million introductions. If the stadium is a lattice grid, the number is so big it would take the universe's lifetime to finish.

In the past, scientists used a trick called the "Two-Point-Source" (2PS) method. They would fix one person in a specific seat and only introduce them to everyone else. This saved time, but it meant they were missing out on the "crowd" statistics. They were only looking at a tiny slice of the city.

The Solution: The "Magic Split"

The authors of this paper realized there was a better way. They found a mathematical "magic trick" that allows them to factorize the problem.

Imagine you are trying to calculate the total noise in a concert hall by listening to every pair of people talking.

• The Old Way: You stand in the middle, pick a person, listen to them talk to everyone else, then move to the next person. It's slow.
• The New Way (Factorization): You realize that the noise a person makes depends only on who they are, not who they are talking to at that exact moment. So, you calculate a "noise profile" for Person A and a "noise profile" for Person B separately. Then, you just multiply these profiles together. Suddenly, you don't need to check every pair; you just combine the lists.

The paper explores three different ways to do this "magic split":

1. The Fourier Method (The Radio Wave):
   This method translates the problem into "radio frequencies" (momentum space). It's like listening to the concert through a radio tuner. It's very fast to set up, but it has a flaw: it doesn't naturally filter out the "static" (noise) from people standing very far away in the stadium. In the simulation, this "static" from the edges of the grid creates a lot of error, especially for the trickier diagrams.

2. The 5D Propagator Method (The 5th Dimension):
   This is the most clever approach. The authors realized that a 4-dimensional photon path can be mathematically split into two 5-dimensional paths that meet in the middle.

   • The Analogy: Imagine two hikers starting at opposite ends of a mountain range. Instead of calculating every step they take together, you calculate the path of Hiker A and the path of Hiker B separately, meeting at a "midpoint" in a 5th dimension.
   • Why it's great: This method naturally includes a "volume knob" that turns down the noise from people standing far away. It's like having a soundproof wall that blocks out the distant chatter. This makes the results much cleaner and more accurate, especially for the difficult parts of the calculation.

3. The Hybrid Approach (The Best of Both Worlds):
   After testing all three methods on a supercomputer (using a grid with 48x48x48 points and a pion mass of 286 MeV), the authors found that no single method was perfect for everything.

   • For the "easy" diagrams, the old Two-Point-Source method was actually the fastest and most accurate because it could reuse data efficiently.
   • For the "hard" diagrams, the 5D Propagator method was the clear winner because it suppressed the noise so well.

The Verdict

The paper concludes that the best strategy is a Hybrid Method.

• Use the Two-Point-Source method for the easy parts.
• Use the 5D Propagator method for the hard parts.

By combining these, they were able to calculate the electromagnetic correction to the muon's wobble with much higher precision than before. Their result is tiny (about -2.85 x 10⁻¹¹), but in the world of particle physics, this tiny number is crucial. It helps refine the Standard Model, bringing us closer to understanding if there is "New Physics" waiting to be discovered.

In short: The authors stopped trying to introduce every person in the stadium to every other person. Instead, they invented a system where they could calculate the "vibe" of each person separately and combine them, saving massive amounts of time and getting a much clearer picture of the universe's secrets.

Technical Summary

1. Problem Statement

The paper addresses the computational challenges associated with calculating electromagnetic (QED) corrections to hadronic vacuum polarization (HVP) in lattice QCD, specifically for the muon anomalous magnetic moment ($a_\mu$).

• The Core Issue: In the position-space approach (specifically the $QED_\infty$ method using Pauli-Villars regularization), the photon propagator depends on the relative coordinates of two internal vertices ($x$ and $y$). This entanglement requires a summation over all pairs of vertices in the lattice volume, leading to a computationally prohibitive volume-squared ($V^2$) sum.
• Previous Limitation: A common workaround (the Two-Point-Source or 2PS method) fixes one vertex to a point source to reduce the sum to $V$. However, this drastically reduces the statistical sampling of the fixed vertex, leading to large errors, particularly for disconnected diagrams and specific topologies like the $(4)b$ diagram.
• Goal: To develop and compare methods that factorize the dependence on the two internal vertices, allowing both volume sums to be performed efficiently without sacrificing statistical precision or incurring excessive computational costs.

2. Methodology

The authors propose and compare three distinct implementations for handling the photon propagator in the calculation of the isospin-violating part of the HVP contribution ($a_\mu^{HVP, 38}$):

A. Two-Point-Source (2PS) Method (Baseline)

• Mechanism: Uses rotational invariance to reduce the integral over one internal vertex ($y$) to a 1D integral along a diagonal. One vertex is fixed at the origin, and the other is sampled along the diagonal.
• Pros: Highly efficient for the connected $(4)a$ diagram due to massive statistical gains (factor of 48 on the $48^3 \times 128$ ensemble) from periodicity.
• Cons: For the $(4)b$ diagram, it requires a sequential propagator that includes the photon propagator. This forces a separate calculation for every position of the $y$-vertex, destroying the statistical advantage and making it a "time sink." It also struggles with disconnected diagrams where noise at large distances is not suppressed.

B. Fourier Method (Factorization via Momentum Space)

• Mechanism: Represents the photon propagator as a superposition of plane waves (momentum eigenstates):
  $$[G(x-y)]_\Lambda = \int d\xi \, \phi_\Lambda(\xi) e^{i\xi \cdot x} e^{-i\xi \cdot y}$$
  Here, $\xi$ represents momentum $k$. The dependence on $x$ and $y$ is factorized into $h^*(x)$ and $g(y)$.
• Pros: Allows access to the unregularized photon propagator (limit $\Lambda \to \infty$) easily; flexible for testing different regularization schemes.
• Cons: The weight functions ($e^{ikx}$) do not decay at large distances. Consequently, noise from large separations is not suppressed, leading to significantly larger errors for disconnected diagrams and UV-finite contributions.

C. 5D Propagator Method (Factorization via Autoconvolution)

• Mechanism: Based on the mathematical identity that a 4D scalar propagator can be expressed as the autoconvolution of a 5D scalar propagator:
  $$G^{(1)}_m(x-y) = \int d\xi \, G^{(3/2)}_m(\xi-x) G^{(3/2)}_m(\xi-y)$$
  The photon propagator is split into a product of two 5D propagators sharing a common "mid-point" $\xi$ (or $w$).
• Pros: The weight functions decay as $1/|x|^3$, naturally suppressing noise from large distances. This is crucial for disconnected diagrams. It also allows for the calculation of unregularized propagators and multiple Pauli-Villars masses efficiently.
• Cons: Higher initial computational cost due to the need to calculate three separate terms (for the PV masses) and the lack of the "periodic diagonal" statistical boost available in the 2PS method.

3. Key Contributions

1. Derivation of Factorization Formulas: The authors derive specific factorization formulas for the photon propagator suitable for both infinite continuous space and infinite lattices (Appendix B), proving the validity of the 5D propagator approach on the lattice.
2. Comprehensive Benchmarking: They perform a rigorous comparison of the three methods on:
   • Gluon-less ensembles: To cross-check against continuum calculations and verify the correct continuum limit.
   • Full QCD Ensemble (N451): A $48^3 \times 128$ lattice with $m_\pi = 286$ MeV, calculating both connected and disconnected diagrams.
3. Hybrid Strategy Proposal: Based on the trade-offs, they propose a hybrid approach: use the 2PS method for the $(4)a$ connected diagram (leveraging its statistical efficiency) and the 5D propagator method for the $(4)b$ connected diagram and all disconnected diagrams (leveraging its noise suppression).

4. Results

• Continuum Limit: All three methods successfully reproduce the expected continuum result for the gluon-less cross-check ($\Lambda = 3m_\mu$), validating the factorization approaches.
• Error Analysis on N451:
  • Connected $(4)a$: The 2PS method yields the smallest errors (factor of $\sim 48$ statistics gain).
  • Connected $(4)b$: The 5D propagator method outperforms the Fourier method. The Fourier method suffers from large errors due to unsuppressed long-range noise. The 5D method provides results consistent with the continuum limit with smaller spreads.
  • Disconnected Diagrams: The 5D propagator method is superior. The Fourier method produces errors orders of magnitude larger than the 5D method because the disconnected loops do not fall off with distance, and the Fourier weights do not suppress this noise. The 2PS method is generally less efficient or inapplicable for certain disconnected topologies without reusing propagators.
• Final Calculation: Using the hybrid strategy on the N451 ensemble, the authors calculate the isospin-violating contribution:
  $$2 a_\mu^{HVP, 38, em} = -2.85(70) \times 10^{-11}$$
  This confirms the extremely small size of $(3+1)$ topology diagrams relative to connected ones.

5. Significance

• Computational Efficiency: The paper provides a practical solution to the $V^2$ bottleneck in lattice QED calculations, enabling precise computations of isospin-breaking effects which are currently a limiting factor in the Standard Model prediction of $a_\mu$.
• Methodological Advancement: The 5D propagator method is identified as a robust technique for handling long-range correlations and disconnected diagrams in QED corrections, offering a significant advantage over Fourier-based approaches in the context of lattice noise.
• Future Applications: The factorization formalism is generalized to suggest applicability to more complex amplitudes, such as the Hadronic Light-by-Light (HLbL) scattering contribution to $a_\mu$, where similar vertex-sum factorizations could reduce computational costs.
• Precision Physics: By reducing the uncertainty in electromagnetic corrections, this work supports the goal of matching the experimental precision of the muon $g-2$ experiments (Fermilab/J-PARC) to rigorously test for New Physics.